= Solution
The <moment-generating function> is $M_X(t)=\mathbb E[e^{tX}]$ at real $t$ for which that expectation is finite; $M_X(0)=1$. For independent <random variables>, the joint law factorizes, so
$$
M_{aX+bY}(t)=\mathbb E[e^{atX}e^{btY}]=\mathbb E[e^{atX}]\mathbb E[e^{btY}]
=\boxed{M_X(at)M_Y(bt)}.
$$
This is the transform form of <independence>. Its ordinary finite-valued interpretation applies where both factors are finite, with the transform of a zero multiple equal to $1$.
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